The equation of a circle is given in general form. Complete parts a through d to follow the step by step process to sketch the graph of the circle. \( x^{2}+y^{2}+4 x-8 y+4=0 \) a) Write the equation of the circle in standard form. The equation is \( (x+2)^{2}+(y-4)^{2}=16 \). (Simplify your answer.) b) Determine the center and radius of the circle. The center is (Type an ordered pair.)

To complete the steps for sketching the graph of the circle given by the equation \( x^{2}+y^{2}+4 x-8 y+4=0 \), let's go through each part. ### a) Write the equation of the circle in standard form. We start with the given equation: \[ x^{2} + y^{2} + 4x - 8y + 4 = 0 \] First, we rearrange it to isolate the constant on one side: \[ x^{2} + y^{2} + 4x - 8y = -4 \] Next, we complete the square for the \(x\) and \(y\) terms. **Completing the square for \(x\):** 1. Take the coefficient of \(x\) (which is 4), halve it (2), and square it (4). 2. Add and subtract this square inside the equation. \[ x^{2} + 4x = (x + 2)^{2} - 4 \] **Completing the square for \(y\):** 1. Take the coefficient of \(y\) (which is -8), halve it (-4), and square it (16). 2. Add and subtract this square inside the equation. \[ y^{2} - 8y = (y - 4)^{2} - 16 \] Now substitute these completed squares back into the equation: \[ (x + 2)^{2} - 4 + (y - 4)^{2} - 16 = -4 \] Combine the constants: \[ (x + 2)^{2} + (y - 4)^{2} - 20 = -4 \] Add 20 to both sides: \[ (x + 2)^{2} + (y - 4)^{2} = 16 \] So, the equation in standard form is: \[ (x + 2)^{2} + (y - 4)^{2} = 16 \] ### b) Determine the center and radius of the circle. From the standard form of the circle's equation \( (x - h)^{2} + (y - k)^{2} = r^{2} \), we can identify: - The center \((h, k)\) is \((-2, 4)\). - The radius \(r\) is \(\sqrt{16} = 4\). Thus, the center is: \[ \text{The center is } (-2, 4). \] ### Summary of Answers: - a) The equation in standard form is \( (x + 2)^{2} + (y - 4)^{2} = 16 \). - b) The center is \((-2, 4)\) and the radius is \(4\).